Problem: All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$2.50$ each for students, and the group paid $$43.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$28.00$ each for teachers and $$9.50$ each for students, and the group paid $$169.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+2.5y = 43}$ ${28x+9.5y = 169}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-28x-10y = -172}$ ${28x+9.5y = 169}$ Add the top and bottom equations together. $ -0.5y = -3 $ $ y = \dfrac{-3}{-0.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7x+2.5y = 43}$ to find $x$ ${7x + 2.5}{(6)}{= 43}$ $7x+15 = 43$ $7x = 28$ $x = \dfrac{28}{7}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {28x+9.5y = 169}$ and get the same answer for $x$ ${28x + 9.5}{(6)}{= 169}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.